for some suitable constant \(c_0>0\) . This shows that \(\Vert \mu \Vert _\ast \le c\)  for all \(\mu \in \Lambda \) . In particular, for \([u,\lambda ]\in X\times \Lambda \)  one has \(\sqrt{\Vert u\Vert ^2+\Vert \lambda \Vert _\ast ^2}\rightarrow \infty \)  if and only if \(\Vert u\Vert \rightarrow \infty \) .
On the other hand, one has (see, e.g., [1]})
\(I(u)\ge \Vert u\Vert ^{\phi ^-}, \forall u\in X, \Vert u\Vert >1,\) 
